Abstract
In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group \(SL(2,{\mathbb {C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-\,1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb {R}})\) and to the Landau operator in the complex plane.
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Notes
There is missed factor \(e^t\) in [6].
\({\mathcal {H}}_y\) is the representation Hilbert space endowed with an inner product \(\langle \cdot , \cdot \rangle _{{\mathcal {H}}_y}\) and \(v_y \in {\mathcal {H}}_y \) is a cyclic SU(2)-invariant unit vector.
References
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, p. xxx+460. Cambridge University Press, Cambridge (2009)
Askour, N., Intissar, A., Mouayn, Z.: Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants, C. R. A. S. Paris, t.325, Série I, pp. 707–712 (1997)
Bernstein, J.N.: On the support of Plancherel measure. J. Geom. Phys. 5(4), 663–710 (1988)
Biane, P.: Intertwining of Markov semi-groups, some examples. Séminaire de probabilités de Strasbourg 29, 30–36 (1995)
Biane, P.: Entrelacements de semi-groupes provenant de paires de Gelfand. ESAIM Probab. Stat. 15, S2–S10 (2011)
Biane, P., Bougerol, P., O’Connell, N.: Continuous crystal and Duistermaat–Heckman measure for Coxeter groups. Adv. Math. 221, 1522–1583 (2009)
Böttcher, B., Schilling, R., Wang, J.: Lévy Matters III. Lévy-type processes: construction, approximation and simple path properties. With a short biography of Paul Lévy by Jean Jacod. Lecture Notes in Mathematics, 2099. Lévy Matters. Springer, Cham (2013)
Boyadzhiev, K.N.: Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of \(\exp (1/x)\). Acta Univ. Sapientiae Math. 8(1), 22–31 (2016)
Carmona, P., Petit, F., Yor, M.: Beta-gamma random variables and intertwining relations between certain Markov processes. Rev. Mat. Iberoam. 14(2), 311–367 (1998)
Chybiryakov, O., Demni, N., Gallardo, L., Rösler, M., Voit, M., Yor, M.: Harmonic and Stochastic Analysis of Dunkl Processes. Travaux en Cours, Hermann (2008)
Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous-elliptiques sur certains groupes nilpotents. Acta Math. 139(1–2), 95–153 (1977)
Gelfand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Representation Theory and Automorphic Functions. Translated from the Russian by K. A. Hirsch. Reprint of the 1969 edition. Generalized Functions, 6. Academic Press, Inc., Boston (1990)
Ghomrasni, R.: On distributions associated with the generalized Lévy’s stochastic area formula. Studia Sci. Math. Hung. 41(1), 93–100 (2004)
Godement, R.: A theory of spherical functions I. Trans. Am. Math. Soc. 73(3), 496–556 (1952)
Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 7th edn. Academic Press, Boston (1994)
Helmes, K.: On Lévy’s Area Process. Stochastic Differential Systems (Bad Honnef, 1985), 187–194, Lecture Notes in Control and Information Science, 78, Springer, Berlin (1986)
Hirsch, F., Yor, M.: Fractional intertwinings between two Markov semigroups. Potential Anal. 31, 133–146 (2009)
Jurek, Z.J., Yor, M., Selfdecomposable laws associated with hyperbolic functions. Probab. Math. Stat. 24 (2004), no. 1. Acta Univ. Wratislav. No. 2646, 181–190 (2004)
Knopova, V., Schilling, R.: A note on the existence of transition probability densities. Forum Math. 25(1), 125–149 (2013)
Koornwinder, T.: Jacobi functions and analysis on non compact semi simple Lie groups. Special Functions: Group Theoretical Aspects and Applications, pp. 1–85, Math. Appl., Reidel, Dordrecht (1984)
Lewandowski, Z., Szynal, J.: An upper bound for the Laguerre polynomials. Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Applications (Seville, 1997). J. Comput. Appl. Math. 99(1–2), 529–533 (1998)
Matsumoto, H., Ueki, N.: Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two dimensional Euclidean space. J. Math. Soc. Jpn. 52(2), 269–292 (2000)
Pitman, J., Yor, M.: A decomposition of Bessel bridges. Z. Wahrsch. Verw. Gebiete. 59(4), 425–457 (1982)
Vershik, A.M., Karpushev, S.I.: Cohomology of groups in unitary representations, the neighborhood of the identity, and conditionally positive definite functions. Math. USSR-Sbornik. 47(2), 513–526 (1984)
Acknowledgements
We would like to thank Jacques Faraut, Philippe Biane, Bachir Bekka and René Schilling for stimulating discussions and remarks.
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Communicated by Hans G. Feichtinger.
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Demni, N. Markov Semi-groups Associated with the Complex Unimodular Group \(Sl(2,{\mathbb {C}})\). J Fourier Anal Appl 25, 2503–2520 (2019). https://doi.org/10.1007/s00041-019-09672-2
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DOI: https://doi.org/10.1007/s00041-019-09672-2
Keywords
- Positive definite functions
- Intertwining operators
- Gelfand pairs
- Metaplectic representation
- Landau Laplacian